Constructions of involutions over finite fields
An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications such as cryptography and coding theory. Following the idea in  to characterize the involutory behavior of the generalized cyclotomic mappings, gives a more concise criterion for $x^rh(x^s)\in \bF_q[x]$ being involutions over the finite field~$\bF_q$, where $r\geq 1$ and $s\,|\, (q-1)$. By using this criterion we propose a general method to construct involutions of the form $x^rh(x^s)$ over $\bF_q$ from given involutions over some subgroups of $\bF_q^*$ by solving congruent and linear equations over finite fields. Then, many classes of explicit involutions of the form $x^rh(x^s)$ over $\bF_q$ are obtained.